zbMATH — the first resource for mathematics

Controlled concentration points and groups of divergence type. (English) Zbl 0872.22007
Johannson, Klaus (ed.), Low-dimensional topology. Proceedings of a conference, held May 18-26, 1992 at the University of Tennessee, Knoxville, TN, USA. Cambridge, MA: International Press. Conf. Proc. Lect. Notes Geom. Topol. 3, 41-45 (1994).
For a nonelementary group \(\Gamma\) of hyperbolic isometries acting on \(B^{d+1}\), the critical exponent \(\delta(\Gamma)\) is defined as the infimum of those \(\alpha\) for which the series \(\Sigma_{\gamma\in\Gamma} e^{-\alpha(0,\gamma(0))}\) converges, where \((0,\gamma(0))\) is the hyperbolic distance from \(0\) to \(\gamma(0)\). The group is said to be of convergence or divergence type according as the series above converges or diverges for \(\alpha=\delta(\Gamma)\). A limit point \(p\in S^d\) of \(\Gamma\) is called a controlled concentration point if it has a neighbourhood \(U\) such that for any connected neighbourhood \(V\) of \(p\) there exists an element \(\gamma\in\Gamma\) such that \(p\in\gamma(V)\) and \(\gamma(U)\subset V\). In the paper under review the author uses results of his thesis to show that for a nonelementary group of divergence type the set of controlled concentration points has full Patterson-Sullivan measure in the set of limit points of \(\Gamma\). In fact, the set of limit points satisfying a stronger condition and which he calls Myrberg-Agard concentration points, has full Patterson-Sullivan measure. This follows from results of the author obtained using S. Agard’s approach [Acta Math. 151, 231-252 (1983; Zbl 0532.30038)] and properties of the Sullivan-Patterson measure.
For the entire collection see [Zbl 0816.00026].

22E40 Discrete subgroups of Lie groups
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)